Homework 2

Generalized Production Function

Part I

Cobb-Douglas: ln(Q) = β₁ + β₂ln(L) + β₃ln(K) + ε
CES: ln(Q) = β₁ + (β₄/β₃)ln(β₂L^β₃ + (1-β₂)K^β₃) + ε

Where Q = VA/NFirm, L=Labor/NFirm, K=Capital/NFirm are per establishment data of value-added, labor, and capital, respectively. Both production functions may be generalized to consider variable rate of returns to scale as follows:

Generalized Cobb-Dougas: ln(Q) + θ Q = β₁ + β₂ln(L) + β₃ln(K) + ε
Generalized CES: ln(Q) + θ Q = β₁ + (β₄/β₃)ln(β₂L^β₃ + (1-β₂)K^β₃) + ε

1. Since the Cobb-Douglas form is a limiting case of CES when the power parameter β₃ approaches 0, estimate and compare the model parameters of Cobb-Douglas and CES production functions. Which functional form is preferred? Explain.

2. Based on the preferred functional form you selected from (1), estimate and interpret the corresponding generalized form to allow variable returns to scale according to Zellner and Revankar [1970]. Is the parameter θ = 0? Explain.

3. The most complicate generalized CES production function may be written as:

   ln(Q) + β₅ Q = β₁ + (β₄/β₃)ln(β₂L^β₃ + (1-β₂)K^β₃) + ε

   Using maximum likelihood method, formulate and estimate the parameter vector β = (β₁, β₂, β₃, β₄, β₅)'. Perform hypothesis testings for β₄ = 1 and β₅ = 0 jointly based on Wald, Lagrangian Multiplier, and Likelihood Ratio tests.

Part II

ln(Q) + θ Q = β₁ + β₂ln(L) + β₃ln(K) + ε

where Q = VA/NFirm, L=Labor/NFirm, K=Capital/NFirm are per establishment data of value-added, labor, and capital, respectively. We assume that the heteroscedastic variances take the following multiplicative form:

σ_i² = σ² h_i(α) and
h_i(α) = exp(Z_iα)

where Z_i = [ln(L_i),ln(K_i)] and α = [α₁,α₂]'. Or, equivalently,

σ_i² = σ² L_i^α1 K_i^α2

Formulate and estimate the log-likelihood function with multiplicative heteroscedasticity. Does the incorporation of heteroscedastic variances improve the model estimates of the generalized Cobb-Douglas production function?